Learning Objectives
Do you play sports like table tennis, badminton or pickled ball?
another example:
Air Defense System - shooting down Fighter Jet
Suppose we have a track cycle of 5 seconds. At intervals of 5 seconds, the radar
samples the target by directing a dedicated pencil beam.
The future target position can be easily calculated using Newton’s motion equations:
Where:
x is the target position
x₀ is the initial target position
v₀ is the initial target velocity
a is the target acceleration
Δt is the time interval (5 seconds in our example)
As for 3D:
The set of target parameters
The 3D motion equations can be written in state-space form:
\begin{bmatrix}
x{k+1} \
y{k+1} \
z{k+1} \
v{x, k+1} \
v{y, k+1} \
v{z, k+1}
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 & \Delta t & 0 & 0 \
0 & 1 & 0 & 0 & \Delta t & 0 \
0 & 0 & 1 & 0 & 0 & \Delta t \
0 & 0 & 0 & 1 & 0 & 0 \
0 & 0 & 0 & 0 & 1 & 0 \
0 & 0 & 0 & 0 & 0 & 1 \
\end{bmatrix}
\begin{bmatrix}
xk \
y_k \
z_k \
v{x,k} \
v{y,k} \
v{z,k}
\end{bmatrix}
Example:
Example:
Some of the values are negative. To get rid of the negative values, let us square the
distance from the mean:
To calculate the variance of the data set, we need to find the average value of all
squared distances from the mean:
-The units of the variance are meters squared; it is more convenient to look at the
standard deviation, which is a square root of the variance.
• The standard deviation of Team B players’ heights would be 0.036m.
Example:
• In city ’A,’ the mean delivery time is 30 minutes, and the standard deviation
is 5 minutes.
• In city ’B,’ the mean delivery time is 40 minutes, and the standard deviation
is 5 minutes.
• In city ’C,’ the mean delivery time is 30 minutes, and the standard deviation
is 10 minutes.
Random Variables
The kth raw moment is:
That is, the expected value (mean) of the kth power of the random variable XXX.
The kth central moment is:
where
This measures how much the values of XXX deviate from the mean, raised to the power